Method of determining torque limit with motor torque and battery power constraints

ABSTRACT

A method to determine a limit torque associated with an electro-mechanical transmission includes determining electric motor torque constraints and battery power constraints. A limit torque function and a standard form of the limit torque function are determined. The limit torque function and the motor torque constraints and the battery power constraints are transposed to the standard form to determine a limit torque.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.60/984,457 filed on Nov. 1, 2007 which is hereby incorporated herein byreference.

TECHNICAL FIELD

This disclosure pertains to control systems for electro-mechanicaltransmissions.

BACKGROUND

The statements in this section merely provide background informationrelated to the present disclosure and may not constitute prior art.

Known hybrid powertrain architectures can include multipletorque-generative devices, including internal combustion engines andnon-combustion machines, e.g., electric machines, which transmit torquethrough a transmission device to an output member. One exemplary hybridpowertrain includes a two-mode, compound-split, electro-mechanicaltransmission which utilizes an input member for receiving tractivetorque from a prime mover power source, preferably an internalcombustion engine, and an output member. The output member can beoperatively connected to a driveline for a motor vehicle fortransmitting tractive torque thereto. Machines, operative as motors orgenerators, can generate torque inputs to the transmission independentlyof a torque input from the internal combustion engine. The machines maytransform vehicle kinetic energy transmitted through the vehicledriveline to energy that is storable in an energy storage device. Acontrol system monitors various inputs from the vehicle and the operatorand provides operational control of the hybrid powertrain, includingcontrolling transmission operating state and gear shifting, controllingthe torque-generative devices, and regulating the power interchangeamong the energy storage device and the machines to manage outputs ofthe transmission, including torque and rotational speed.

SUMMARY

A powertrain includes an electro-mechanical transmissionmechanically-operatively coupled to an internal combustion engine andfirst and second electric machines to transfer power to an outputmember. A method to determine a limit torque associated with theelectro-mechanical transmission includes monitoring operation of theelectro-mechanical transmission, determining motor torque constraints ofthe first and second electric machines and battery power constraints forthe electrical energy storage device. A limit torque function isdetermined and an original form of the limit torque function identified.The limit torque function is transposed to a standard form, and themotor torque constraints and the battery power constraints aretransposed. A transposed limit torque in the standard form is determinedbased upon the transposed limit torque function, the transposed motortorque constraints, and the transposed battery power constraints. Thetransposed limit torque is retransposed from the standard form to theoriginal form to determine the limit torque in the original form.

BRIEF DESCRIPTION OF THE DRAWINGS

One or more embodiments will now be described, by way of example, withreference to the accompanying drawings, in which:

FIG. 1 is a schematic diagram of an exemplary powertrain, in accordancewith the present disclosure; and,

FIG. 2 is a schematic diagram of an exemplary architecture for a controlsystem and powertrain, in accordance with the present disclosure; and

FIGS. 3-9 are graphical depictions, in accordance with the disclosure.

DETAILED DESCRIPTION

Referring now to the drawings, wherein the showings are for the purposeof illustrating certain exemplary embodiments only and not for thepurpose of limiting the same, FIGS. 1 and 2 depict an exemplaryelectro-mechanical hybrid powertrain. The exemplary electro-mechanicalhybrid powertrain in accordance with the present disclosure is depictedin FIG. 1, comprising a two-mode, compound-split, electro-mechanicalhybrid transmission 10 operatively connected to an engine 14 and firstand second electric machines (‘MG-A’) 56 and (‘MG-B’) 72. The engine 14and first and second electric machines 56 and 72 each generate powerwhich can be transferred to the transmission 10. The power generated bythe engine 14 and the first and second electric machines 56 and 72 andtransferred to the transmission 10 is described in terms of input andmotor torques, referred to herein as T_(I), T_(A), and T_(B)respectively, and speed, referred to herein as N_(I), N_(A), and N_(B),respectively.

The exemplary engine 14 comprises a multi-cylinder internal combustionengine selectively operative in several states to transfer torque to thetransmission 10 via an input shaft 12, and can be either aspark-ignition or a compression-ignition engine. The engine 14 includesa crankshaft (not shown) operatively coupled to the input shaft 12 ofthe transmission 10. A rotational speed sensor 11 monitors rotationalspeed of the input shaft 12. Power output from the engine 14, comprisingrotational speed and engine torque, can differ from the input speedN_(I) and the input torque T_(I) to the transmission 10 due to placementof torque-consuming components on the input shaft 12 between the engine14 and the transmission 10, e.g., a hydraulic pump (not shown) and/or atorque management device (not shown).

The exemplary transmission 10 comprises three planetary-gear sets 24, 26and 28, and four selectively engageable torque-transferring devices,i.e., clutches C1 70, C2 62, C3 73, and C4 75. As used herein, clutchesrefer to any type of friction torque transfer device including single orcompound plate clutches or packs, band clutches, and brakes, forexample. A hydraulic control circuit 42, preferably controlled by atransmission control module (hereafter ‘TCM’) 17, is operative tocontrol clutch states. Clutches C2 62 and C4 75 preferably comprisehydraulically-applied rotating friction clutches. Clutches C1 70 and C373 preferably comprise hydraulically-controlled stationary devices thatcan be selectively grounded to a transmission case 68. Each of theclutches C1 70, C2 62, C3 73, and C4 75 is preferably hydraulicallyapplied, selectively receiving pressurized hydraulic fluid via thehydraulic control circuit 42.

The first and second electric machines 56 and 72 preferably comprisethree-phase AC machines, each including a stator (not shown) and a rotor(not shown), and respective resolvers 80 and 82. The motor stator foreach machine is grounded to an outer portion of the transmission case68, and includes a stator core with coiled electrical windings extendingtherefrom. The rotor for the first electric machine 56 is supported on ahub plate gear that is operatively attached to shaft 60 via the secondplanetary gear set 26. The rotor for the second electric machine 72 isfixedly attached to a sleeve shaft hub 66.

Each of the resolvers 80 and 82 preferably comprises a variablereluctance device including a resolver stator (not shown) and a resolverrotor (not shown). The resolvers 80 and 82 are appropriately positionedand assembled on respective ones of the first and second electricmachines 56 and 72. Stators of respective ones of the resolvers 80 and82 are operatively connected to one of the stators for the first andsecond electric machines 56 and 72. The resolver rotors are operativelyconnected to the rotor for the corresponding first and second electricmachines 56 and 72. Each of the resolvers 80 and 82 is signally andoperatively connected to a transmission power inverter control module(hereafter ‘TPIM’) 19, and each senses and monitors rotational positionof the resolver rotor relative to the resolver stator, thus monitoringrotational position of respective ones of first and second electricmachines 56 and 72. Additionally, the signals output from the resolvers80 and 82 are interpreted to provide the rotational speeds for first andsecond electric machines 56 and 72, i.e., N_(A) and N_(B), respectively.

The transmission 10 includes an output member 64, e.g. a shaft, which isoperably connected to a driveline 90 for a vehicle (not shown), toprovide output power to the driveline 90 that is transferred to vehiclewheels 93, one of which is shown in FIG. 1. The output power at theoutput member 64 is characterized in terms of an output rotational speedN_(O) and an output torque T_(O). A transmission output speed sensor 84monitors rotational speed and rotational direction of the output member64. Each of the vehicle wheels 93 is preferably equipped with a sensor94 adapted to monitor wheel speed, the output of which is monitored by acontrol module of a distributed control module system described withrespect to FIG. 2, to determine vehicle speed, and absolute and relativewheel speeds for braking control, traction control, and vehicleacceleration management.

The input torque from the engine 14 and the motor torques from the firstand second electric machines 56 and 72 (T_(I), T_(A), and T_(B)respectively) are generated as a result of energy conversion from fuelor electrical potential stored in an electrical energy storage device(hereafter ‘ESD’) 74. The ESD 74 is high voltage DC-coupled to the TPIM19 via DC transfer conductors 27. The transfer conductors 27 include acontactor switch 38. When the contactor switch 38 is closed, undernormal operation, electric current can flow between the ESD 74 and theTPIM 19. When the contactor switch 38 is opened electric current flowbetween the ESD 74 and the TPIM 19 is interrupted. The TPIM 19 transmitselectrical power to and from the first electric machine 56 by transferconductors 29, and the TPIM 19 similarly transmits electrical power toand from the second electric machine 72 by transfer conductors 31 tomeet the torque commands for the first and second electric machines 56and 72 in response to the motor torques T_(A) and T_(B). Electricalcurrent is transmitted to and from the ESD 74 in accordance with whetherthe ESD 74 is being charged or discharged.

The TPIM 19 includes the pair of power inverters (not shown) andrespective motor control modules (not shown) configured to receive thetorque commands and control inverter states therefrom for providingmotor drive or regeneration functionality to meet the commanded motortorques T_(A) and T_(B). The power inverters comprise knowncomplementary three-phase power electronics devices, and each includes aplurality of insulated gate bipolar transistors (not shown) forconverting DC power from the ESD 74 to AC power for powering respectiveones of the first and second electric machines 56 and 72, by switchingat high frequencies. The insulated gate bipolar transistors form aswitch mode power supply configured to receive control commands. Thereis typically one pair of insulated gate bipolar transistors for eachphase of each of the three-phase electric machines. States of theinsulated gate bipolar transistors are controlled to provide motor drivemechanical power generation or electric power regenerationfunctionality. The three-phase inverters receive or supply DC electricpower via DC transfer conductors 27 and transform it to or fromthree-phase AC power, which is conducted to or from the first and secondelectric machines 56 and 72 for operation as motors or generators viatransfer conductors 29 and 31 respectively.

FIG. 2 is a schematic block diagram of the distributed control modulesystem. The elements described hereinafter comprise a subset of anoverall vehicle control architecture, and provide coordinated systemcontrol of the exemplary hybrid powertrain described in FIG. 1. Thedistributed control module system synthesizes pertinent information andinputs, and executes algorithms to control various actuators to meetcontrol objectives, including objectives related to fuel economy,emissions, performance, drivability, and protection of hardware,including batteries of ESD 74 and the first and second electric machines56 and 72. The distributed control module system includes an enginecontrol module (hereafter ‘ECM’) 23, the TCM 17, a battery pack controlmodule (hereafter ‘BPCM’) 21, and the TPIM 19. A hybrid control module(hereafter ‘HCP’) 5 provides supervisory control and coordination of theECM 23, the TCM 17, the BPCM 21, and the TPIM 19. A user interface(‘UI’) 13 is signally connected to a plurality of devices through whicha vehicle operator controls or directs operation of theelectro-mechanical hybrid powertrain. The devices include an acceleratorpedal 113 (‘AP’), an operator brake pedal 112 (‘BP’), a transmissiongear selector 114 (‘PRNDL’), and a vehicle speed cruise control (notshown). The transmission gear selector 114 may have a discrete number ofoperator-selectable positions, including the rotational direction of theoutput member 64 to enable one of a forward and a reverse direction.

The aforementioned control modules communicate with other controlmodules, sensors, and actuators via a local area network (hereafter‘LAN’) bus 6. The LAN bus 6 allows for structured communication ofstates of operating parameters and actuator command signals between thevarious control modules. The specific communication protocol utilized isapplication-specific. The LAN bus 6 and appropriate protocols providefor robust messaging and multi-control module interfacing between theaforementioned control modules, and other control modules providingfunctionality including e.g., antilock braking, traction control, andvehicle stability. Multiple communications buses may be used to improvecommunications speed and provide some level of signal redundancy andintegrity. Communication between individual control modules can also beeffected using a direct link, e.g., a serial peripheral interface(‘SPI’) bus (not shown).

The HCP 5 provides supervisory control of the hybrid powertrain, servingto coordinate operation of the ECM 23, TCM 17, TPIM 19, and BPCM 21.Based upon various input signals from the user interface 13 and thehybrid powertrain, including the ESD 74, the HCP 5 determines anoperator torque request, an output torque command, an engine inputtorque command, clutch torque(s) for the applied torque-transferclutches C1 70, C2 62, C3 73, C4 75 of the transmission 10, and themotor torques T_(A) and T_(B) for the first and second electric machines56 and 72. The TCM 17 is operatively connected to the hydraulic controlcircuit 42 and provides various functions including monitoring variouspressure sensing devices (not shown) and generating and communicatingcontrol signals to various solenoids (not shown) thereby controllingpressure switches and control valves contained within the hydrauliccontrol circuit 42.

The ECM 23 is operatively connected to the engine 14, and functions toacquire data from sensors and control actuators of the engine 14 over aplurality of discrete lines, shown for simplicity as an aggregatebi-directional interface cable 35. The ECM 23 receives the engine inputtorque command from the HCP 5. The ECM 23 determines the actual engineinput torque, T_(I), provided to the transmission 10 at that point intime based upon monitored engine speed and load, which is communicatedto the HCP 5. The ECM 23 monitors input from the rotational speed sensor11 to determine the engine input speed to the input shaft 12, whichtranslates to the transmission input speed, N_(I). The ECM 23 monitorsinputs from sensors (not shown) to determine states of other engineoperating parameters including, e.g., a manifold pressure, enginecoolant temperature, ambient air temperature, and ambient pressure. Theengine load can be determined, for example, from the manifold pressure,or alternatively, from monitoring operator input to the acceleratorpedal 113. The ECM 23 generates and communicates command signals tocontrol engine actuators, including, e.g., fuel injectors, ignitionmodules, and throttle control modules, none of which are shown.

The TCM 17 is operatively connected to the transmission 10 and monitorsinputs from sensors (not shown) to determine states of transmissionoperating parameters. The TCM 17 generates and communicates commandsignals to control the transmission 10, including controlling thehydraulic circuit 42. Inputs from the TCM 17 to the HCP 5 includeestimated clutch torques for each of the clutches, i.e., C1 70, C2 62,C3 73, and C4 75, and rotational output speed, N_(O), of the outputmember 64. Other actuators and sensors may be used to provide additionalinformation from the TCM 17 to the HCP 5 for control purposes. The TCM17 monitors inputs from pressure switches (not shown) and selectivelyactuates pressure control solenoids (not shown) and shift solenoids (notshown) of the hydraulic circuit 42 to selectively actuate the variousclutches C1 70, C2 62, C3 73, and C4 75 to achieve various transmissionoperating range states, as described hereinbelow.

The BPCM 21 is signally connected to sensors (not shown) to monitor theESD 74, including states of electrical current and voltage parameters,to provide information indicative of parametric states of the batteriesof the ESD 74 to the HCP 5. The parametric states of the batteriespreferably include battery state-of-charge, battery voltage, batterytemperature, and available battery power, referred to as a range P_(BAT)_(—) _(MIN) to P_(BAT) _(—) _(MAX).

A brake control module (hereafter ‘BrCM’) 22 is operatively connected tofriction brakes (not shown) on each of the vehicle wheels 93. The BrCM22 monitors the operator input to the brake pedal 112 and generatescontrol signals to control the friction brakes and sends a controlsignal to the HCP 5 to operate the first and second electric machines 56and 72 based thereon.

Each of the control modules ECM 23, TCM 17, TPIM 19, BPCM 21, and BrCM22 is preferably a general-purpose digital computer comprising amicroprocessor or central processing unit, storage mediums comprisingread only memory (‘ROM’), random access memory (‘RAM’), electricallyprogrammable read only memory (‘EPROM’), a high speed clock, analog todigital (‘A/D’) and digital to analog (‘D/A’) circuitry, andinput/output circuitry and devices (‘I/O’) and appropriate signalconditioning and buffer circuitry. Each of the control modules has a setof control algorithms, comprising resident program instructions andcalibrations stored in one of the storage mediums and executed toprovide the respective functions of each computer. Information transferbetween the control modules is preferably accomplished using the LAN bus6 and serial peripheral interface buses. The control algorithms areexecuted during preset loop cycles such that each algorithm is executedat least once each loop cycle. Algorithms stored in the non-volatilememory devices are executed by one of the central processing units tomonitor inputs from the sensing devices and execute control anddiagnostic routines to control operation of the actuators, using presetcalibrations. Loop cycles are executed at regular intervals, for exampleeach 3.125, 6.25, 12.5, 25 and 100 milliseconds during ongoing operationof the hybrid powertrain. Alternatively, algorithms may be executed inresponse to the occurrence of an event.

The exemplary hybrid powertrain selectively operates in one of severaloperating range states that can be described in terms of an engine statecomprising one of an engine-on state (‘ON’) and an engine-off state(‘OFF’), and a transmission state comprising a plurality of fixed gearsand continuously variable operating modes, described with reference toTable 1, below.

TABLE 1 Engine Transmission Operating Applied Description State RangeState Clutches M1_Eng_Off OFF EVT Mode 1 C1 70 M1_Eng_On ON EVT Mode 1C1 70 G1 ON Fixed Gear Ratio 1 C1 70 C4 75 G2 ON Fixed Gear Ratio 2 C170 C2 62 M2_Eng_Off OFF EVT Mode 2 C2 62 M2_Eng_On ON EVT Mode 2 C2 62G3 ON Fixed Gear Ratio 3 C2 62 C4 75 G4 ON Fixed Gear Ratio 4 C2 62 C373

Each of the transmission operating range states is described in thetable and indicates which of the specific clutches C1 70, C2 62, C3 73,and C4 75 are applied for each of the operating range states. A firstcontinuously variable mode, i.e., EVT Mode 1, or M1, is selected byapplying clutch C1 70 only in order to “ground” the outer gear member ofthe third planetary gear set 28. The engine state can be one of ON(‘M1_Eng_On’) or OFF (‘M1_Eng_Off’). A second continuously variablemode, i.e., EVT Mode 2, or M2, is selected by applying clutch C2 62 onlyto connect the shaft 60 to the carrier of the third planetary gear set28. The engine state can be one of ON (‘M2_Eng_On’) or OFF(‘M2_Eng_Off’). For purposes of this description, when the engine stateis OFF, the engine input speed is equal to zero revolutions per minute(‘RPM’), i.e., the engine crankshaft is not rotating. A fixed gearoperation provides a fixed ratio operation of input-to-output speed ofthe transmission 10, i.e., N_(I)/N_(O). A first fixed gear operation(‘G1’) is selected by applying clutches C1 70 and C4 75. A second fixedgear operation (‘G2’) is selected by applying clutches C1 70 and C2 62.A third fixed gear operation (‘G3’) is selected by applying clutches C262 and C4 75. A fourth fixed gear operation (‘G4’) is selected byapplying clutches C2 62 and C3 73. The fixed ratio operation ofinput-to-output speed increases with increased fixed gear operation dueto decreased gear ratios in the planetary gears 24, 26, and 28. Therotational speeds of the first and second electric machines 56 and 72,N_(A) and N_(B) respectively, are dependent on internal rotation of themechanism as defined by the clutching and are proportional to the inputspeed measured at the input shaft 12.

In response to operator input via the accelerator pedal 113 and brakepedal 112 as captured by the user interface 13, the HCP 5 and one ormore of the other control modules determine torque commands to controlthe torque generative devices comprising the engine 14 and first andsecond electric machines 56 and 72 to meet the operator torque requestat the output member 64 and transferred to the driveline 90. Based uponinput signals from the user interface 13 and the hybrid powertrainincluding the ESD 74, the HCP 5 determines the operator torque request,a commanded output torque between the transmission 10 and the driveline90, an input torque from the engine 14, clutch torques for thetorque-transfer clutches C1 70, C2 62, C3 73, C4 75 of the transmission10; and the motor torques for the first and second electric machines 56and 72, respectively, as is described hereinbelow. The commanded outputtorque can be a tractive torque wherein torque flow originates in theengine 14 and the first and second electric machines 56 and 72 and istransferred through the transmission 10 to the driveline 90, and can bea reactive torque wherein torque flow originates in the vehicle wheels93 of the driveline 90 and is transferred through the transmission 10 tofirst and second electric machines 56 and 72 and the engine 14.

Final vehicle acceleration can be affected by other factors including,e.g., road load, road grade, and vehicle mass. The operating range stateis determined for the transmission 10 based upon a variety of operatingcharacteristics of the hybrid powertrain. This includes the operatortorque request communicated through the accelerator pedal 113 and brakepedal 112 to the user interface 13 as previously described. Theoperating range state may be predicated on a hybrid powertrain torquedemand caused by a command to operate the first and second electricmachines 56 and 72 in an electrical energy generating mode or in atorque generating mode. The operating range state can be determined byan optimization algorithm or routine which determines optimum systemefficiency based upon operator demand for power, battery state ofcharge, and energy efficiencies of the engine 14 and the first andsecond electric machines 56 and 72. The control system manages torqueinputs from the engine 14 and the first and second electric machines 56and 72 based upon an outcome of the executed optimization routine, andsystem efficiencies are optimized thereby, to manage fuel economy andbattery charging. Furthermore, operation can be determined based upon afault in a component or system. The HCP 5 monitors the torque-generativedevices, and determines the power output from the transmission 10required in response to the desired output torque at output member 64 tomeet the operator torque request. As should be apparent from thedescription above, the ESD 74 and the first and second electric machines56 and 72 are electrically-operatively coupled for power flowtherebetween. Furthermore, the engine 14, the first and second electricmachines 56 and 72, and the electro-mechanical transmission 10 aremechanically-operatively coupled to transfer power therebetween togenerate a power flow to the output member 64.

Operation of the engine 14 and transmission 10 is constrained by power,torque and speed limits of the engine 14, the first and second electricmachines 56 and 72, the ESD 74 and the clutches C1 70, C2 62, C3 73, andC4 75. The operating constraints on the engine 14 and transmission 10can be translated to a set of system constraint equations executed asone or more algorithms in one of the control modules, e.g., the HCP 5.

Referring again to FIG. 1, in overall operation, the transmission 10operates in one of the operating range states through selectiveactuation of one or two of the torque-transfer clutches. Torqueconstraints for each of the engine 14 and the first and second electricmachines 56 and 72 and speed constraints for each of the engine 14, thefirst and second electric machines 56 and 72, and the output shaft 64 ofthe transmission 10 are determined. Battery power constraints for theESD 74 are determined, and are applied to further limit the motor torqueconstraints for the first and second electrical machines 56 and 72. Thepreferred operating region for the powertrain is determined using thesystem constraint equation, based upon the battery power constraints,the motor torque constraints, and the speed constraints. The preferredoperating region comprises a range of permissible operating torques orspeeds for the engine 14 and the first and second electric machines 56and 72.

By deriving and simultaneously solving dynamics equations of thetransmission 10, the torque limit, in this embodiment the output torqueT_(O), can be determined using the following linear equations:T _(M1) =T _(A) to T _(M1) *T _(A) +T _(B) to T _(M1) *T _(B)+Misc_(—) T_(M1)  [1]T _(M2) =T _(A) to T _(M2) *T _(A) +T _(B) to T _(M2) *T _(B)+Misc_(—) T_(M2)  [2]T _(M3) =T _(A) to T _(M3) *T _(A) +T _(B) to T _(M3) *T _(B)+Misc_(—) T_(M3)  [3]wherein, in this embodiment,

-   -   T_(M1) represents the output torque T_(O) at output member 64,    -   T_(M2) represents the input torque T_(I) at input shaft 12,    -   T_(M3) represents the reactive clutch torque(s) for the applied        torque-transfer clutches C1 70, C2 62, C3 73, C4 75 of the        transmission 10,    -   T_(A) to T_(M1), T_(A) to T_(M2), T_(A) to T_(M3) are        contributing factors of T_(A) to T_(M1), T_(M2), T_(M3),        respectively,    -   T_(B) to T_(M1), T_(B) to T_(M2), T_(B) to T_(M3) are        contributing factors of T_(B) to T_(M1), T_(M2), T_(M3),        respectively,    -   Misc_T_(M1), Misc_T_(M2), and Misc_T_(M3) are constants which        contribute to T_(M1), T_(M2), T_(M3) by N_(I) _(—) _(DOT), N_(O)        _(—) _(DOT), and N_(C) _(—) _(DOT) (time-rate changes in the        input speed, output speed and clutch slip speed) respectively,        and    -   T_(A) and T_(B) are the motor torques from the first and second        electric machines 56 and 72.        The torque parameters T_(M1), T_(M2), T_(M3) can be any three        independent parameters, depending upon the application.

The engine 14 and transmission 10 and the first and second electricmachines 56 and 72 have speed constraints, torque constraints, andbattery power constraints due to mechanical and system limitations.

The speed constraints can include engine speed constraints of N_(I)=0(engine off state), and N_(I) ranging from 600 rpm (idle) to 6000 rpmfor the engine 14. The speed constraints for the first and secondelectric machines 56 and 72 can be as follows:−10,500 rpm≦N_(A)≦+10,500 rpm, and−10,500 rpm≦N_(B)≦+10,500 rpm.

The torque constraints include engine torque constraints including T_(I)_(—) _(MIN)<T_(I)<T_(I) _(—) _(MAX), and motor torque constraints forthe first and second electric machines including T_(A) _(—)_(MIN)<T_(A)<T_(A) _(—) _(MAX) and T_(B) _(—) _(MIN)<T_(B)<T_(B) _(—)_(MAX). The motor torque constraints T_(A) _(—) _(MAX) and T_(A) _(—)_(MIN) comprise torque limits for the first electric machine 56 whenworking as a torque-generative motor and an electrical generator,respectively. The motor torque constraints T_(B) _(—) _(MAX) and T_(B)_(—) _(MIN) comprise torque limits for the second electric machine 72when working as a torque-generative motor and an electrical generator,respectively. The maximum and minimum motor torque constraints T_(A)MAX, T_(A) MIN, T_(B) _(—) _(MAX), T_(B) _(—) _(MIN) are preferablyobtained from data sets stored in tabular format within one of thememory devices of one of the control modules. Such data sets areempirically derived from conventional dynamometer testing of thecombined motor and power electronics (e.g., power inverter) at varioustemperature and voltage conditions.

Battery power constraints comprise the available battery power withinthe range of P_(BAT) _(—) _(MIN) to P_(BAT) _(—) _(MAX), wherein P_(BAT)_(—) _(MIN) is maximum allowable battery charging power and P_(BAT) _(—)_(MAX) is the maximum allowable battery discharging power. Battery poweris defined as positive when discharging and negative when charging.

Minimum and maximum values for T_(M1) are determined within the speedconstraints, the motor torque constraints, clutch torque constraints,and the battery power constraints during ongoing operation, in order tocontrol operation of the engine 14, the first and second electricmachines 56 and 72, also referred to hereinafter as Motor A 56 and MotorB 72, and the transmission 10 to meet the operator torque request andthe commanded output torque.

An operating range, comprising a torque output range is determinablebased upon the battery power constraints of the FSD 74. Calculation ofbattery power usage, P_(BAT) is as follows:P _(BAT) =P _(A,ELEC) +P _(B,ELEC) +P _(DC) _(—) _(LOAD)  [4]wherein P_(A,ELEC) comprises electrical power from Motor A 56,

-   -   P_(B,ELEC) comprises electrical power from Motor B 72, and    -   P_(DC) _(—) _(LOAD) comprises known DC load, including accessory        loads.

Substituting equations for P_(A,ELEC) and P_(B,ELEC), yields thefollowing:P _(BAT)=(P _(A,MECH) +P _(A,LOSS))+(P _(B,MECH) +P _(B,LOSS))+P _(DC)_(—) _(LOAD)  [5]wherein P_(A,MECH) comprises mechanical power from Motor A 56,

-   -   P_(A,LOSS) comprises power losses from Motor A 56,    -   P_(B,MECH) comprises mechanical power from Motor B 72, and    -   P_(B,LOSS) comprises power losses from Motor B 72.

Eq. 5 can be restated as Eq. 6, below, wherein speeds, N_(A) and N_(B),and torques, T_(A) and T_(B), are substituted for powers P_(A) andP_(B). This includes an assumption that motor and inverter losses can bemathematically modeled as a quadratic equation based upon torque asfollows:

$\begin{matrix}{P_{BAT} = {\begin{pmatrix}{{N_{A}T_{A}} +} \\\begin{pmatrix}{{{a_{1}\left( N_{A} \right)}T_{A}^{2}} +} \\{{{a_{2}\left( N_{A} \right)}T_{A}} + {a_{3}\left( N_{A} \right)}}\end{pmatrix}\end{pmatrix} + \begin{pmatrix}{{N_{B}T_{B}} +} \\\begin{pmatrix}{{{b_{1}\left( N_{B} \right)}T_{B}^{2}} +} \\{{{b_{2}\left( N_{B} \right)}T_{B}} + {b_{3}\left( N_{B} \right)}}\end{pmatrix}\end{pmatrix} + P_{{DC}_{\_ LOAD}}}} & \lbrack 6\rbrack\end{matrix}$wherein N_(A), N_(B) comprise speeds of Motors A and B 56 and 72,

-   -   T_(A), T_(B) comprise torques of Motors A and B 56 and 72, and    -   a1, a2, a3, b1, b2, b3 each comprise quadratic coefficients        which are a function of respective motor speeds, N_(A), N_(B).

This can be restated as Eq. 7 as follows.

$\begin{matrix}{P_{BAT} = {{a_{1}*T_{A}^{2}} + {\left( {N_{A} + a_{2}} \right)*T_{A}} + {b_{1}*T_{B}^{2}} + {\left( {N_{B} + b_{2}} \right)*T_{B}} + {a\; 3} + {b\; 3} + P_{DC\_ LOAD}}} & \lbrack 7\rbrack\end{matrix}$

This reduces to Eq. 8 as follows.

$\begin{matrix}{P_{BAT} = {{a_{1}\begin{bmatrix}{T_{A}^{2} + {T_{A}\frac{\left( {N_{A} + a_{2}} \right)}{a_{1}}} +} \\\left( \frac{\left( {N_{A} + a_{2}} \right)}{\left( {2*a_{1}} \right)} \right)^{2}\end{bmatrix}} + {b_{1}\begin{bmatrix}{T_{B}^{2} + {T_{B}\frac{\left( {N_{B} + b_{2}} \right)}{b_{1}}} +} \\\left( \frac{\left( {N_{B} + b_{2}} \right)}{\left( {2*b_{1}} \right)} \right)^{2}\end{bmatrix}} + {a\; 3} + {b\; 3} + P_{DC\_ LOAD} - \frac{\left( {N_{A} + a_{2}} \right)^{2}}{\left( {4*a_{1}} \right)} - \frac{\left( {N_{B} + b_{2}} \right)^{2}}{\left( {4*b_{1}} \right)}}} & \lbrack 8\rbrack\end{matrix}$

This reduces to Eq. 9 as follows.P _(BAT) =a ₁ [T _(A)+(N _(A) +a ₂)/(2*a ₁)]² +b ₁ [T _(B)+(N _(B) +b₂)/(2*b ₁)]² +a ₃ +b ₃ +P _(DC) _(—) _(LOAD)−(N _(A) +a ₂)²/(4*a ₁)−(N_(B) +b ₂)²/(4*b ₁)  [9]

This reduces to Eq. 10 as follows.

$\begin{matrix}{P_{BAT} = {\begin{bmatrix}{{{SQRT}\left( a_{1} \right)*T_{A}} +} \\\frac{\left( {N_{A} + a_{2}} \right)}{\left( {2*{{SQRT}\left( a_{1} \right)}} \right)}\end{bmatrix}^{2} + {\begin{bmatrix}{{{SQRT}\left( b_{1} \right)*T_{B}} +} \\\frac{\left( {N_{B} + b_{2}} \right)}{\left( {2*{{SQRT}\left( b_{1} \right)}} \right)}\end{bmatrix}2} + a_{3} + b_{3} + P_{DC\_ LOAD} - \frac{\left( {N_{A} + a_{2}} \right)^{2}}{\left( {4*a_{1}} \right)} - \frac{\left( {N_{B} + b_{2}} \right)^{2}}{\left( {4*b_{1}} \right)}}} & \lbrack 10\rbrack\end{matrix}$

This reduces to Eq. 11 as follows.P _(BAT)=(A ₁ *T _(A) +A ₂)²+(B ₁ *T _(B) +B ₂)² +C  [11]wherein A₁=SQRT(a₁),

-   -   B₁=SQRT(b₁),    -   A₂=(N_(A)+a₂)/(2*SQRT(a₁)),    -   B₂=(N_(B)+b₂)/(2*SQRT(b₁)), and    -   C=a₃+b₃+P_(DC) _(—)        _(LOAD)−(N_(A)+a₂)²/(4*a₁)−(N_(B)+b₂)²/(4*b₁)

The motor torques T_(A) and T_(B) can be transformed to T_(X) and T_(Y)as follows:

$\begin{matrix}{\begin{bmatrix}T_{X} \\T_{Y}\end{bmatrix} = {{\begin{bmatrix}A_{1} & 0 \\0 & B_{1}\end{bmatrix}*\begin{bmatrix}T_{A} \\T_{B}\end{bmatrix}} + \begin{bmatrix}A_{2} \\B_{2}\end{bmatrix}}} & \lbrack 12\rbrack\end{matrix}$wherein T_(X) is the transformation of T_(A),

-   -   T_(Y) is the transformation of T_(B), and    -   A₁, A₂, B₁, B₂ comprise application-specific scalar values.

Eq. 11 can thus be further reduced as follows.P _(BAT)=(T _(X) ² +T _(Y) ²)+C  [13]P _(BAT) =R ² +C  [14]

Eq. 12 specifies the transformation of motor torque T_(A) to T_(X) andthe transformation of motor torque T_(B) to T_(Y). Thus, a newcoordinate system referred to as T_(X)/T_(Y) space is defined, and Eq.13 comprises battery power, P_(BAT), transformed into T_(X)/T_(Y) space.Therefore, the battery power range between maximum and minimum batterypower P_(BAT) _(—) _(MAX) and P_(BAT) _(—) _(MIN) can be calculated andgraphed as radii R_(Max) and R_(Min) with a center at locus (0, 0) inthe transformed space T_(X)/T_(Y), designated by the letter K as shownwith reference to FIG. 3, wherein:R _(Min)=SQRT(P _(BAT) _(—) _(MIN) −C), andR _(Max)=SQRT(P _(BAT) _(—) _(MAX) −C).

The minimum and maximum battery powers, P_(BAT) _(—) _(MIN) and P_(BAT)_(—) _(MAX), are preferably correlated to battery physics, e.g. state ofcharge, temperature, voltage and usage (amp-hour/hour). The parameter C,above, is defined as the absolute minimum possible battery power atgiven motor speeds, N_(A) and N_(B), within the motor torque limits.Physically, when T_(A)=0 and T_(B)=0 the output power from the first andsecond electric machines 56 and 72 is zero. Physically T_(X)=0 andT_(Y)=0 corresponds to a maximum charging power condition for the ESD74. The positive sign (‘+’) is defined as discharging power from the ESD74, and the negative sign (‘−’) is defined as charging power into theESD 74. R_(Max) defines a maximum battery power, typically a dischargingpower, and R_(Min) defines a maximum battery power.

The forgoing transformations to the T_(X)/T_(Y) space are shown in FIG.3, with representations of the battery power constraints as concentriccircles having radii of R_(Min) and R_(Max) (‘Battery PowerConstraints’) and linear representations of the motor torque constraints(‘Motor Torque Constraints’) circumscribing an allowable operatingregion. Analytically, the transformed vector [T_(X) T_(Y)] determined inEq. 12 is solved simultaneously with the vector defined in Eq. 13comprising the minimum and maximum battery powers identified by R_(Min)and R_(Max) to identify a range of allowable torques in the T_(X)/T_(Y)space which are made up of motor torques T_(A) and T_(B) constrained bythe minimum and maximum battery powers P_(BAT) _(—) _(MIN) to P_(BAT)_(—) _(MAX). The range of allowable torques in the T_(X)/T_(Y) space isshown with reference to FIG. 3, wherein points A, B, C, D, and Erepresent the bounds, and lines and radii are defined.

A constant torque line can be defined in the T_(X)/T_(Y) space, anddepicted in FIG. 3 (‘T_(M1)=C1’), comprising the limit torque T_(M1),described in Eq. 1, above. The limit torque T_(M1) comprises the outputtorque T_(O) in this embodiment, Eqs. 1, 2, and 3 restated in theT_(X)/T_(Y) space are as follows.T _(M1) =T _(A) to T_(M1)*(T _(X) −A ₂)/A ₁ +T _(B) to T _(M1)*(T _(Y)−B ₂)/B ₁+Misc_(—) T _(M2)  [15]T _(M2) =T _(A) to T _(M2)*(T _(X) −A ₂)/A ₁ +T _(B) to T _(M2)*(T _(Y)−B ₂)/B ₁+Misc_(—) T _(M2)  [16]T _(M3) =T _(A) to T _(M3)*(T _(X) −A ₂)/A ₁ +T _(B) to T _(M3)*(T _(Y)−B ₂)/B ₁+Misc_(—) T _(M3)  [17]

Defining T_(M1) _(—) _(XY), T_(M2) _(—) _(XY), and T_(M3) _(—) _(XY) asparts of T_(M1), T_(M2), and T_(M3), contributed by T_(A) and T_(B)only, then:T _(M1) _(—) _(XY) =T _(A) to T _(M1)*(T _(X) −A ₂)/A ₁ +T _(B) to T_(M1)*(T _(Y) −B ₂)/B ₁  [18]T _(M2) _(—) _(XY) =T _(A) to T _(M2)*(T _(X) −A ₂)/A ₁ +T _(B) to T_(M2)*(T _(Y) −B ₂)/B ₁  [19]T _(M3) _(—) _(XY) =T _(A) to T _(M3)*(T _(X) −A ₂)/A ₁ +T _(B) to T_(M3)*(T _(Y) −B ₂)/B ₁  [20]

The following coefficients can be defined:T _(X) to T _(M1) =T _(A) to T _(M1) /A ₁,T _(Y) to T _(M1) =T _(B) to T _(M1) /B ₁,T _(M1) _(—) Intercept=T _(A) to T _(M1) *A ₂ /A ₁ +T _(B) to T _(M1) *B₂ /B ₁,T _(X) to T _(M2) =T _(A) to T _(M2) /A ₁,T _(Y) to T _(M2) =T _(B) to T _(M2) /B ₁,T _(M2) _(—) Intercept=T _(A) to T _(M2) *A ₂ /A ₁ +T _(B) to T _(M2) *B₂ /B ₁,T _(X) to T _(M3) =T _(A) to T _(M3) /A ₁,T _(Y) to T _(M3) =T _(B) to T _(M3) /B ₁, andT _(M3) _(—) Intercept=T _(A) to T _(M3) *A ₂ /A ₁ +T _(B) to T_(M3) *B₂ /B ₁.

Thus, Eqs. 1, 2, and 3 are transformed to T_(X)/T_(Y) space as follows.T _(M1) _(—) _(XY) =T _(X) to T _(M1) *T _(X) +T _(Y) to T _(M1) *T _(Y)+T _(M1) _(—) Intercept  [21]T _(M2) _(—) _(XY) =T _(X) to T _(M2) *T _(X) +T _(Y) to T _(M2) *T _(Y)+T _(M2) _(—) Intercept  [22]T _(M3) _(—) _(XY) =T _(X) to T _(M3) *T _(X) +T _(Y) to T _(M3) *T _(Y)+T _(M3) _(—) Intercept  [23]

The speed constraints, motor torque constraints, and battery powerconstraints can be determined during ongoing operation and expressed inlinear equations which are transformed to T_(X)/T_(Y) space. Eq. 21comprises a limit torque function describing the output torqueconstraint T_(M1), e.g., T_(O). The limit torque function can besimultaneously solved with the speed constraints, motor torqueconstraints, and battery power constraints to determine a transformedmaximum or minimum limit torque in the T_(X)/T_(Y) space, comprising oneof T_(M1) _(—) _(XY)Max and T_(M1) _(—) _(XY)Min, i.e., maximum andminimum output torques T_(O) _(—) _(Max) and T_(O) _(—) _(Min) that havebeen transformed. Subsequently the transformed maximum or minimum limittorque in the T_(X)/T_(Y) space can be retransformed out of theT_(X)/T_(Y) space to determine maximum or minimum limit torques T_(M1)_(—) _(Max) and T_(M1) _(—) _(Min) for managing control and operation ofthe transmission 14 and the first and second electric machines 56 and72.

FIGS. 4A and 4B graphically show a first configuration including astandard form of the limit torque function defined in Eq. 21, which isshown as Line A of FIG. 4A. The standard form of the limit torquefunction comprises a combination of a minimum motor torque T_(A)transformed to T_(X) and a maximum motor torque T_(B) transformed toT_(Y) with the coefficients T_(X) to T_(M1) and T_(Y) to T_(M1) havingarithmetic signs as shown in the standard equation as follows.T _(M1) =−T _(X) +T _(Y)(‘Tm1=−Tx+Ty’)  [24]

FIGS. 4A and 4B further show the transformed motor torque constraintsand transformed battery power constraints. The transformed motor torqueconstraints and transformed battery power constraints associated withLine A comprise the standard forms for the transformed motor torqueconstraints and transformed battery power constraints. The transformedlimit torque for T_(M1) is a maximum limit torque, comprising a maximumoutput torque, i.e., a maximum tractive torque achievable within themotor torque constraints and battery power constraints.

Line B of FIG. 4A shows a second form of the limit torque functiondefined in Eq. 21, with the coefficients T_(X) to T_(M1) and T_(Y) toT_(M1) having arithmetic signs as follows.T _(M1) =T _(X) −T _(Y)(‘Tm1=Tx−Ty’)  [25]

Line C of FIG. 4B shows a third form of the limit torque functiondefined in Eq. 21, with the coefficients T_(X) to T_(M1) and T_(Y) toT_(M1) having arithmetic signs as follows.T _(M1) =−T _(X) −T _(Y)(‘Tm1=−Tx−Ty’)  [26]

Line D of FIG. 4B shows a fourth form of the limit torque functiondefined in Eq. 21, with the coefficients T_(X) to T_(M1) and T_(Y) toT_(M1) having arithmetic signs as shown.T _(M1) =T _(X) +T _(Y)(‘Tm1=Tx+Ty’)  [27]

The maximum limit torques in FIGS. 4A and 4B are shown as T_(M1) _(—)Max(A), T_(M1) _(—) Max(B), T_(M1) _(—) _(Max(C)), and T_(M1) _(—)Max(D), graphically representing minimum limit torques for theconstraint equations. The arithmetic signs of the coefficients T_(X) toT_(M1) and T_(Y) to T_(M1), i.e., positive and negative, are indicativeof charging or discharging actions by the respective first and secondelectric machines 56 and 72

FIGS. 5A and 5B graphically show the motor torque constraints andbattery power constraints. The limit torque for T_(M1) is a minimumlimit torque, comprising a minimum output torque, i.e., a maximumreactive torque achievable within the motor torque constraints and thebattery power constraints.

Line A of FIG. 5A shows the fifth form of the limit torque functiondefined in Eq. 21, comprising a minimum output torque with thecoefficients T_(X) to T_(M1) and T_(Y) to T_(M1) having arithmetic signsas shown in the standard equation as follows.T _(M1) =−T _(X) +T _(Y)(‘Tm1=−Tx+Ty’)  [28]

Line B of FIG. 5A shows a sixth form of the limit torque functiondefined in Eq. 21, with the coefficients T_(X) to T_(M1) and T_(Y) toT_(M1) having arithmetic signs as follows.T _(M1) =T _(X) −T _(Y)(‘Tm1=Tx−Ty’)  [29]

Line C of FIG. 5B shows a seventh form of the limit torque functiondefined in Eq. 21, with the coefficients T_(X) to T_(M1) and T_(Y) toT_(M1) having arithmetic signs as follows.T _(M1) =−T _(X) −T _(Y)(‘Tm1=−Tx−Ty’)  [30]

Line D of FIG. 5B shows an eighth form of the limit torque functiondefined in Eq. 21, with the coefficients T_(X) to T_(M1) and T_(Y) toT_(M1) having arithmetic signs as follows.T _(M1) =T _(X) +T _(Y)(‘Tm1=Tx+Ty’)  [31]

The minimum limit torques in FIGS. 5A and 5B are shown as T_(M1) _(—)Min(A), T_(M1) _(—) Min(B), T_(M1) _(—) Min(C), and T_(M1) _(—) Min(D),graphically representing minimum limit torques for the constraintequations. The arithmetic signs of the coefficients T_(X) to T_(M1) andT_(Y) to T_(M1), i.e., positive and negative, are indicative of chargingor discharging actions by the respective first and second electricmachines 56 and 72.

Advantageously, the transformation of the constraint equations toT_(X)/T_(Y) space described hereinabove is reduced to a singleexecutable algorithm that simultaneously solves Eqs. 12, 14, and 21 todetermine one of the maximum limit torque and the minimum limit torqueduring ongoing operation of the transmission 10, the first and secondelectric machines 56 and 72 and the engine 14 to meet the operatortorque request and the torque command as constrained by the motor torqueconstraints and the battery power constraints. The algorithm ispreferably executed during ongoing operation at least once during each25 msec loop cycle for controlling operation of the transmission 14.

Furthermore, the eight combinations of limit equations described abovewith reference to FIGS. 4A, 4B, 5A, and 5B can each be adapted toutilize the single executable algorithm that simultaneously solves Eqs.12, 14, and 21 to determine one of the maximum limit torque and theminimum limit torque. This includes executing the algorithm aftertransposing the electric motor torque constraints and battery powerconstraints to a standard form. This includes monitoring operation anddetermining electric motor torque constraints and battery powerconstraints. The standard limit torque function and an original form ofthe limit torque function are determined. The limit torque function, themotor torque constraints and the battery power constraints aretransposed from the original form to the standard form. A transposedlimit torque state is determined in the standard form based upon thetransposed limit torque function and the transposed motor torqueconstraints and the transposed battery power constraints by executingthe executable algorithm. The transposed torque limit state determinedin the standard form is retransposed back to the original form todetermine the limit torque state in the original form. This is nowdescribed in detail.

FIG. 6 graphically illustrates determining the maximum limit torqueT_(M1) _(—) Max(A) when the original form of the limit torque functionof Eq. 21 is T_(M1)=−T_(X)+T_(Y). The maximum limit torque T_(M1) _(—)Max(A), i.e., Point P, is determined by executing the algorithm whichcalculates intersection(s) of the determined motor torque constraints(‘Motor Torque Constraints’), the battery power constraints (‘BatteryPower Constraints’) and the limit torque function (‘A’). The algorithmis executed in real time to determine the maximum value T_(M1) _(—)Max(A) based upon the constraints. The original form of theconfiguration of FIG. 6 is designated as the standard form. Since theoriginal form of the limit torque function is the standard form, thereis no need to transpose and retranspose to an original form.

FIGS. 7A-7E graphically illustrate determining the maximum limit torqueT_(M1) _(—) Max(B) when the original form of the limit torque function(‘B’) of Eq. 21 is T_(M1)=T_(X)−T_(Y). The maximum value T_(M1) _(—)Max(B) is determined by executing the algorithm which calculatesintersections of the determined motor torque constraints (‘Motor TorqueConstraints’), the battery power constraints (‘Battery PowerConstraints’) and the limit torque function (‘B’), shown in FIG. 7A. Thelimit torque function (‘B’) is transposed symmetrically about both theT_(X) axis and the T_(Y) axis compared to the standard form describedwith reference to Line A of FIG. 4A. Therefore, the motor torqueconstraints can be transposed about both the T_(X) axis and the T_(Y)axis, creating transposed motor torque constraints (‘Transposed MotorTorque Constraints’) that are consistent with the transposed limittorque function (“B”), as shown in FIG. 7B. FIG. 7C graphically showsthe transposed limit torque function (“B”) which is transposed aboutboth the T_(X) axis and the T_(Y) axis. Subsequent to transposing aboutboth the T_(X) axis and the T_(Y) axis into the standard form, thealgorithm is executed to identify the transposed maximum value (‘Q’)based upon the battery power constraints and the transposed motor torqueconstraints for the transposed limit torque function. FIG. 7D shows thetransposed maximum value (‘Q’) retransposed about both the T_(X) axisand the T_(Y) axis to the original form to identify the maximum value(‘R’) corresponding to the maximum limit torque T_(M1) _(—) Max(B),i.e., the maximum limit torque in the original form. FIG. 7E validatesthat the maximum value (‘R’) corresponding to the maximum limit torqueT_(M1) _(—) Max(B) also corresponds to a value on the motor torqueconstraints.

FIGS. 8A-8E graphically illustrate determining the maximum limit torqueT_(M1) _(—) Max(C) when the original form of the limit torque function(‘C’) of Eq. 21 is T_(M1)=−T_(X)−T_(Y), as shown in FIG. 8A. The maximumvalue T_(M1) _(—) Max(C) is determined by executing the algorithm whichcalculates intersections of the determined motor torque constraints(‘Motor Torque Constraints’), the battery power constraints (‘BatteryPower Constraints’) and the limit torque function (‘C’). The limittorque function (‘C’) of Eq. 21 is transposed symmetrically about theT_(X) axis compared to the standard form described with reference toLine A of FIG. 4A. Therefore, the motor torque constraints can betransposed about the T_(X) axis, creating transposed motor torqueconstraints (‘Transposed Motor Torque Constraints’) that are consistentwith the transposed limit torque function (“C”), as shown in FIG. 8B.FIG. 8C graphically shows the transposed limit torque function (“C”) ofEq. 21 transposed about the T_(X) axis. Subsequent to transposing aboutthe T_(X) axis to the standard form, the algorithm is executed toidentify the transposed maximum limit torque (‘Q’) within the batterypower constraints and the transposed motor torque constraints. FIG. 8Dshows the transposed maximum value (‘Q’) is retransposed about the T_(X)axis to identify the maximum value (‘R’) corresponding to the maximumvalue T_(M1) _(—) Max(C), i.e., the maximum limit torque in the originalform. FIG. 8E validates that the maximum value (‘R’) corresponding tothe maximum limit torque T_(M1) _(—) Max(C) corresponds to a value onthe torque constraints.

FIGS. 9A-9E graphically illustrate determination of the maximum valueT_(M1) _(—) Max(D) when the original form of the limit torque function(‘D’) of Eq. 21 is T_(M1)=T_(X)+T_(Y), as shown in FIG. 9A. The maximumvalue T_(M1) _(—) Max(D) is determined by executing the algorithm whichcalculates based upon intersections of the determined motor torqueconstraints (‘Motor Torque Constraints’), the battery power constraints(‘Battery Power Constraints’) and the limit torque function (‘D’). Thelimit torque function (‘D’) of Eq. 21 is transposed symmetrically aboutthe T_(Y) axis compared to the standard form described with reference toLine A of FIG. 4A. Therefore, the motor torque constraints can betransposed about the T_(Y) axis, creating transposed motor torqueconstraints (‘Transposed Motor Torque Constraints’) that are consistentwith the transposed limit torque function (“D”), as shown in FIG. 9B.FIG. 9C graphically shows the transposed limit torque function (“D”) ofEq. 21 transposed about the T_(Y) axis. Subsequent to transposing aboutthe T_(Y) axis, the algorithm is executed to identify the transposedmaximum value (‘Q’) within the battery power constraints and thetransposed motor torque constraints. FIG. 9D shows the transposedmaximum value (‘Q’) is retransposed about the T_(Y) axis to identify themaximum value (‘R’) corresponding to the maximum value T_(M1) _(—)Max(D), i.e., the maximum limit torque in the original form. FIG. 9Evalidates that the maximum value (‘R’) corresponding to the maximumlimit torque T_(M1) _(—) Max(D) corresponds to a value on the torqueconstraints.

Similarly, the minimum limit torques T_(M1) _(—) Min(A), T_(M1) _(—)Min(B), T_(M1) _(—) Min(C) and T_(M1) _(—) Min(D) can be determinedusing the same sign conventions and symmetry about one or both the T_(X)axis and the T_(Y) axis by transposing the original form of the limittorque function, determining the transposed limit torque, andretransposing to determine the minimum limit torque, i.e., the maximumreactive torque.

The maximum or minimum transformed limit torque for motor torques T_(X)and T_(Y) are retransformed to motor torques T_(A) and T_(B) as follows.

$\begin{matrix}{\begin{bmatrix}T_{A} \\T_{B}\end{bmatrix} = {{\begin{bmatrix}{{1/A}\; 1} & 0 \\0 & {{1/B}\; 1}\end{bmatrix}*\begin{bmatrix}T_{X} \\T_{Y}\end{bmatrix}} + \begin{bmatrix}{{- A}\;{2/A}\; 1} \\{{- B}\;{2/B}\; 2}\end{bmatrix}}} & \lbrack 32\rbrack\end{matrix}$

The control system controls operation of the first and second electricmachines 56 and 72 based upon the motor torques T_(A) and T_(B).Executing the equations as algorithms in T_(X)/T_(Y) space decreasesexecution time to a single controller loop cycle, which is the 12.5millisecond cycle in this application.

It is understood that modifications are allowable within the scope ofthe disclosure. The disclosure has been described with specificreference to the preferred embodiments and modifications thereto.Further modifications and alterations may occur to others upon readingand understanding the specification. It is intended to include all suchmodifications and alterations insofar as they come within the scope ofthe disclosure.

1. Method to determine a limit torque associated with anelectro-mechanical transmission operatively coupled to first and secondelectric machines electrically connected to an electrical energy storagedevice to transfer mechanical power to an output member, comprising:monitoring operation of the electro-mechanical transmission; determiningmotor torque constraints of the first and second electric machines andbattery power constraints for the electrical energy storage device;determining a limit torque function and identifying an original form ofthe limit torque function; transposing the limit torque function to astandard form; transposing the motor torque constraints and the batterypower constraints; determining a transposed limit torque in the standardform based upon the transposed limit torque function and the transposedmotor torque constraints and the transposed battery power constraints;and retransposing the transposed limit torque from the standard form tothe original form to determine the limit torque in the original form. 2.The method of claim 1, further comprising: transforming the motor torqueconstraints of the first and second electric machines and the batterypower constraints and the limit torque function antecedent totransposing the limit torque function to the standard form,retransforming the determined limit torque in the original form, andcontrolling the first and second electric machines based upon theretransformed limit torque.
 3. The method of claim 1, further comprisingcontrolling operation of the electro-mechanical transmission and thefirst and second electric machines based upon the limit torque.
 4. Themethod of claim 3, wherein the limit torque comprises an output torqueto the output member.
 5. The method of claim 3, further comprising theelectro-mechanical transmission including a selectively applicabletorque transfer clutch and the limit torque comprising a reactive torqueof the selectively applied torque transfer clutch.
 6. The method ofclaim 3, further comprising the electro-mechanical transmissionincluding an input member and the limit torque comprising an inputtorque to the input member.
 7. The method of claim 2, comprisingdetermining the transformed limit torque by simultaneously solving thetransformed limit torque function, transformed motor torque constraints,and transformed battery power constraints to determine one of a maximumand a minimum transformed limit torque.
 8. The method of claim 1,wherein the limit torque comprises a minimum output torque to the outputmember achievable within the motor torque constraints and battery powerconstraints.
 9. The method of claim 8, wherein the minimum output torqueto the output member comprises a maximum regenerative torque output forelectric power generation.
 10. The method of claim 1, wherein the limittorque comprises a maximum output torque to the output member achievablewithin the motor torque constraints and the battery power constraints.11. The method of claim 10, wherein the maximum output torque to theoutput member comprises a maximum mechanical torque output for tractivepower generation.
 12. The method of claim 1, wherein the standard formcomprises a combination of a minimum torque output from the firstelectric machine and a maximum torque output from the second electricmachine.
 13. The method of claim 1, further comprising transposing themotor torque constraints and the battery power constraints consistentwith the transposing the limit torque function to the standard form. 14.Method to determine a limit torque associated with an electro-mechanicaltransmission, operatively coupled to first and second electric machineselectrically connected to an electrical energy storage device totransfer mechanical power to an output member, comprising: monitoringoperation of the electro-mechanical transmission; determining motortorque constraints of the first and second electric machines and batterypower constraints for the electrical energy storage device; determininga limit torque function and identifying an original form of the limittorque function; transforming the motor torque constraints of the firstand second electric machines, the battery power constraints, and thelimit torque function to a TX/TY space; transposing the transformedlimit torque function to a standard form comprising a combination of aminimum torque output from the first electric machine and a maximumtorque output from the second electric machine; transposing thetransformed motor torque constraints and the transformed battery powerconstraints; determining a transposed transformed limit torque in thestandard form based upon the transposed transformed limit torquefunction and the transposed transformed motor torque constraints and thetransposed transformed battery power constraints; retransposing thetransposed transformed limit torque from the standard form to theoriginal form to determine the transformed limit torque in the originalform; and retransforming the transformed limit torque in the originalform.
 15. The method of claim 14, further comprising controllingoperation of the electro-mechanical transmission and the first andsecond electric machines based upon the retransformed limit torque. 16.The method of claim 15, wherein the limit torque comprises an outputtorque to the output member.
 17. The method of claim 14, furthercomprising the electro-mechanical transmission including a selectivelyapplicable torque transfer clutch and the limit torque comprising areactive torque of the selectively applied torque transfer clutch. 18.The method of claim 14, further comprising the electro-mechanicaltransmission including an input member and the limit torque comprisingan input torque to the input member.
 19. The method of claim 14,comprising determining the transformed limit torque by simultaneouslysolving the transformed limit torque function, transformed motor torqueconstraints, and transformed battery power constraints to determine oneof a maximum and a minimum transformed limit torque.